Biochemistry - The Chemical Reactions of Living Cells Volume 2 - D. Metzler 1980

Enzymes: Protein Catalysts of Cells
Fundamentals of Enzyme Kinetics
The "Cell Effect" and Molecular Rotation

It is interesting to compare the second-order rate constant for productive collisions, calculated using Smoluchowski's theory [i.e., via equation (6-26)], with the second-order rate constant A for molecular collisions derived from the kinetic theory of gases:

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1) Personal communication from D. French, to whom the author is deeply grateful for discussing the issues concerning Collision Theory. Parts of these Structure/133.html">Discussion results were previously published [18].

In this expression, m_A and m_B are the masses of particles A and B, respectively, and k_B is the Boltzmann constant. The rate constant k also exhibits a relatively weak dependence on molecular size, ranging from 4∙10¹¹ to 11∙10¹¹ M^-1∙s^-1 for spherical particles—which is slightly more than an order of magnitude larger than the rate constant for productive collisions. Gas molecules collide and immediately bounce off one another. In solution, however, they continue to collide with a nearly constant frequency after the initial encounter, resulting in 100 to 200 collisions between the two particles for every productive collision. During a single productive collision, the particles remain together within a solvent "cage"; this feature plays a crucial role in enzymatic reactions.

During a productive collision between a substrate and an enzyme, both molecules undergo random rotational motions, so successive collisions catch them in varying relative orientations. In one of such orientations, the complementary surfaces (i.e., the substrate and the substrate-binding site) come into close proximity, leading to The formation of a "productive" ES complex.

Molecular rotation in solution is quantitatively described by a diffusion law (analogous to Fick's law), which incorporates the rotational diffusion coefficient 0 [19, 20]. Let us consider a group of molecules whose initial orientations are identical and subsequently undergo random changes due to rotational diffusion. The orientation of each molecule can be characterized by the angle a:

At the initial moment, cos a = 1, and as the orientation changes randomly, the average value of cos a, obtained by averaging over all molecules, approaches zero. The rotational relaxation time t is defined as the time required for the average value of cos a to decrease by a factor of e (at which point a becomes equal to 68.5°). For spherical molecules, the following relation holds:

Molecules shaped as ellipsoids of revolution or rods are characterized by two rotational diffusion coefficients, whereas molecules with dimensions differing along all three axes possess three.

Suppose that the diameter of a small spherical substrate molecule is 1 nm, while that of a spherical enzyme molecule is 5 nm. The θ values for these molecules at η = 0.01 P, calculated using equation (6-32), are respectively

As we can see, smaller molecules rotate much faster than large ones, and the rotational relaxation times of small Proteins are of the same order of magnitude as the rate constant kВ for diffusion-controlled productive collisions. However, for very large molecules (especially long, rod-shaped ones), the rotational relaxation time around the short axis can reach fractions of a second.